Question: Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (5 \mathbf{v}).$
Answer: From the formula for a projection,
\begin{align*}
\operatorname{proj}_{\mathbf{w}} (5 \mathbf{v}) &= \frac{(5 \mathbf{v}) \cdot \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w} \\
&= \frac{5 \mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w} \\
&= 5 \operatorname{proj}_{\mathbf{w}} \mathbf{v} \\
&= 5 \begin{pmatrix} 3 \\ 2 \end{pmatrix} \\
&= \boxed{\begin{pmatrix} 15 \\ 10 \end{pmatrix}}.
\end{align*}